CN113037663A - Improved code element rate estimation algorithm suitable for non-constant envelope signal - Google Patents
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Abstract
The invention relates to an improved code element rate estimation algorithm suitable for non-constant envelope, which aims at the problem that the code element rate estimation algorithm based on the envelope square spectrum fails in MASK signals and signals formed by low roll-off factors. The improved algorithm solves the problem that the original algorithm has large estimation error in MASK signals and low roll-off factor forming signals, has wider signal application range than the original algorithm, has low complexity and is beneficial to engineering realization.
Description
Technical Field
The invention relates to an improved code element rate estimation algorithm suitable for a non-constant envelope signal, belonging to the technical field of digital communication.
Background
The symbol rate, also called symbol rate, is one of the main parameters for distinguishing analog signals from digital signals, and is also an important parameter for characterizing signals. The accurate estimation of the symbol rate has important application value in the fields of radio frequency spectrum management and monitoring, signal identification and demodulation and the like, and from the aspect of identification of a modulation mode in uncooperative communication with less prior knowledge, the accurate estimation of the symbol rate plays a prerequisite role in recovery of a digital modulation symbol, so the accurate estimation of the symbol rate is very important in preprocessing of signals. In addition, after the modulation scheme is identified, a high-accuracy symbol rate is also required for blind demodulation of the signal. Therefore, the symbol rate estimation has high research value.
At present, the main methods for estimating the code element rate mainly include methods based on a cyclic spectrum, wavelet transformation, an envelope square spectrum and the like, wherein the method based on the cyclic spectrum has higher complexity and is not beneficial to engineering realization; the wavelet transform-based method requires a high signal-to-noise ratio and requires selection of an appropriate wavelet transform to obtain a good estimation effect. The envelope square spectrum has the advantages of simple principle, low calculation complexity and high robustness, and is suitable for engineering application.
The code element rate estimation algorithm based on the envelope square spectrum is mainly applied to MQAM signals and is suitable for MPSK and MQAM signals under the condition of large roll-off factor formation, but the algorithm fails in MASK signals, and meanwhile, a large number of experiments show that the algorithm has larger estimation error for MPSK and MQAM signals under the condition of small roll-off factor formation, so that the method has the problem of sensitivity to roll-off factors and modulation types. In actual life, especially in the military field, a shaping filter with a small roll-off factor is often used for shaping to save the frequency spectrum and improve the utilization rate of the frequency spectrum, and the range of the roll-off factor commonly used is [0.15,0.5 ]. Therefore, the invention provides an improved symbol rate estimation algorithm suitable for a non-constant envelope signal.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides an improved code element rate estimation algorithm suitable for a non-constant envelope signal, which is based on an envelope square spectrum and aims to solve the problem that the algorithm fails in MASK signals and signals formed by low roll-off factors, and the improved algorithm is provided, so that the improved algorithm is suitable for the non-constant envelope signal, the application range is expanded, the characteristic of low complexity is considered, and a solution is provided for engineering realization.
Interpretation of terms:
1. fast Fourier transform: the fast algorithm of discrete Fourier transform is obtained by improving the algorithm of discrete Fourier transform according to the characteristics of odd, even, imaginary and real of the discrete Fourier transform.
2. Normalization: the data is adjusted to a specified standard form. In the present invention, each data point is divided by the maximum value of the sequence, so that the maximum value of the value is 1, and the others are less than 1.
3. Removing direct current: the spectral values around 0 are set to zero.
4. Roll-off factor: also known as roll-off coefficient, is the ratio of the required bandwidth to the symbol transmission rate (i.e., the nyquist frequency) without intersymbol interference.
MASK: multiple amplitude-shift keying, short for multiple amplitude-shift keying, is a multilevel digital amplitude modulation.
MPSK: multiple phase shift keying, short for multiple phase shift keying, is a multilevel digital phase modulation.
MQAM: short for Multiple Quadrature Amplitude Modulation, multilevel Quadrature Amplitude Modulation.
8. Normalized root mean square error: the Normalized root mean square error, NRMSE for short, is a statistical value obtained by normalizing the root mean square difference, and is a commonly used measure of the difference between measured values, which is often a quantity predicted by a model or an observed estimated quantity.
Zoomft: referred to as a refined fast fourier transform, also known as a bandlet fast fourier transform. The function of the zoomft is to perform local refinement and amplification on the frequency of the signal, so that the frequency band of interest obtains higher frequency resolution.
10. Linear modulation Z transform (CZT): is an algorithm suitable for calculating when the reciprocal of the product of the sampling frequency interval and the sampling time interval is not equal to the time-frequency distribution area of the signal, which is a fast fourier transform algorithm that implements Discrete Fourier Transform (DFT) of an arbitrary size using convolution.
The technical scheme of the invention is as follows:
an improved symbol rate estimation algorithm for a non-constant envelope signal for estimating the symbol rate of the non-constant envelope signal, the method comprising the steps of:
According to the invention, the envelope square spectrum of the signal is calculated, the relative saliency is calculated for the amplitude of each frequency point, the relative saliency is multiplied by the original amplitude to correct the envelope square spectrum, the maximum value of the corrected spectrum is extracted to obtain the rough estimation, and the frequency selection analysis is adopted to carry out the fine estimation, so that the estimated value of the code element rate is obtained. The improved algorithm solves the problem that the original algorithm has large estimation error in MASK signals and low roll-off factor forming signals, has wider signal application range than the original algorithm, has low complexity and is beneficial to engineering realization.
Preferably, in step 1, the received signal sequence r (t) is obtained through an additive white gaussian noise channel, and the expression of r (t) is shown in formula (I):
r(t)＝s(t)+n(t) (I)
in the formula (I), n (t) represents a mean value of 0 and a variance of
White gaussian noise independent of the signal s (t), s (t) is expressed by the formula (II):
in the formula (II), x (t) represents a complex baseband symbol sequence after passing through a shaping filter, fcThe carrier frequency, x (t), is expressed as in formula (III):
in the formula (III), cn＝an+jbnRepresents a modulation symbol, anRepresenting baseband modulation symbolsReal part of the sign, bnRepresenting the imaginary part of the baseband modulation symbols, n representing the number of baseband modulation symbols, gT(T) represents a function of the shaping filter, TbIs the symbol period.
Preferably, according to the invention, the shaping filter is a root-raised cosine roll-off shaping filter. The root raised cosine roll-off shaping filter is selected to save frequency spectrum resources.
Preferably, in step 3, R isp(f) Is defined in the present invention as the relative projection, Rp(f) The calculation formula of (2) is as follows:
in the formula (IV), | U (f) | represents the amplitude value of the spectral line at the frequency f, | Ul(f)|、|Ur(f) I respectively represents the maximum value of the spectral line amplitude of the left k neighborhood of the frequency f; i Ur(f) I respectively represents the maximum value of the spectral line amplitude of the right k neighborhood of the frequency f; the left k neighborhood represents the frequency point range of [ f-k, f-1 ]]The right k neighborhood represents the frequency point range of [ f +1, f + k ]]The frequency spectrum of (a);
further preferably, k is 32.
For lines in the continuum, their relative prominence R is due to their close proximity to the surrounding lines in magnitudepApproximately equal to 1, and the more prominent spectral lines are much larger than their surrounding spectral lines, with relative prominence RpFar greater than 1, the evaluation criteria presented herein can effectively evaluate the prominence of a spectral line with respect to its surrounding spectral lines. The extraction of the parameters can not only effectively depict the prominence of the code element rate spectral line, but also has better application significance in the modulation mode identification based on the spectral characteristics, and can numerically depict the spectral characteristics of signals.
Preferably, in step 5, in a narrow-band range, a simplified zoomft is used to perform frequency-selecting analysis on { a (t) } to obtain a frequency spectrum m (f), and the specific steps are as follows:
5-1, selecting a thinning multiple D, wherein the size of D determines the estimation accuracy, the larger D is, the higher the estimation accuracy is, but the corresponding calculated amount is increased, and the D is 32 or 64 in the relation of the balance accuracy and the calculated amount;
5-2, mixing U+In the case of' (F) f.epsilon. [ F ]coa,(1+ε)Fcoa]The corresponding data is arranged and moved to the U in the forward direction+To the right of the zero frequency position of' (f), similarly, U+' (F) wherein F is within [ (1-epsilon) Fcoa,Fcoa) The corresponding data is arranged and moved to the U in the forward direction+Left side of zero frequency position of' (f), setting other position frequency spectrum data to zero to obtain new sequence S (f);
5-3, performing N/2-point inverse Fourier transform on the S (f) to obtain a time domain signal, wherein the value of N is more than or equal to the minimum power number of 2 of the length of the { A (t) } sequence;
5-4, resampling the time domain signal, wherein the sampling frequency of resampling is fs/D，fsRepresenting sampling frequency, namely taking data every other (D-1) points to obtain m (t), wherein m (t) represents a time domain signal after resampling;
5-5, performing N/2-point Fourier transform on m (t) to obtain a refined frequency spectrum M (f).
In the case of a fixed sampling rate, the number of sampling points of the signal needs to be increased to obtain higher estimation accuracy, which inevitably increases the amount of calculation. However, in practical engineering, the smaller the number of data points required for fast estimation, the better. Therefore, in order to solve the contradiction between the precision and the calculated amount, a frequency-selective analysis algorithm is required to be adopted to carry out fine estimation on the narrow band. Compared with the common CZT algorithm, the simplified ZoomFFT reduces multiple times of exponential operation, reduces exponential and filtering operation compared with the traditional ZoomFFT algorithm, has a frequency selection analysis function, and greatly reduces algorithm complexity.
The invention has the beneficial effects that:
1. the improved algorithm provided by the invention is used for estimating the code element rate of the non-constant envelope signal, solves the problem of large estimation error of an envelope square spectrum on MASK signals and low roll-off factor molding signals, and has wider signal application range compared with the original algorithm.
2. The improved algorithm provided by the invention aims at {2ASK, 4ASK, 8ASK, BPSK and Q under the condition of not filtering in advancePSK, 8PSK, 16PSK, pi/4 DQPS, 8QAM, 16QAM, 32QAM, 64QAM }12 kinds of non-constant envelope modulation signals, when the signal-to-noise ratio is more than 4dB, the Normalized Root Mean Square Error (NRMSE) is reduced to 10-4The following are excellent in noise immunity.
3. When the sampling rate is fixed, the higher estimation precision can be obtained only by increasing the number of data points, which inevitably increases the calculation amount.
4. The improved algorithm provided by the invention does not need to carry out any preprocessing such as carrier synchronization, code element synchronization and the like on the received signal, has low complexity, is beneficial to engineering realization and has extremely high engineering realization significance.
Drawings
FIG. 1 is a flow chart of an improved symbol rate estimation algorithm for a non-constant envelope signal according to the present invention;
FIG. 2(a) is a 2ASK signal envelope square spectrogram obtained based on an unmodified algorithm;
fig. 2(b) is a 2ASK signal envelope square spectrogram obtained by the improved symbol rate estimation algorithm applied to the non-constant envelope signal provided in embodiment 1;
fig. 2(c) is a frequency spectrum of an improved symbol rate estimation algorithm applied to a non-constant envelope signal, which is provided in embodiment 1, after being refined in step 5;
FIG. 3(a) is a diagram of the NRMSE change in MASK class of the improved symbol rate estimation algorithm for non-constant envelope signals provided in example 1;
fig. 3(b) is a diagram of NRMSE variation in MPSK and MQAM classes of an improved symbol rate estimation algorithm suitable for non-constant envelope signals provided in embodiment 1;
FIG. 4 is a graph showing the variation of NRMSE in MASK-like signals by a symbol rate estimation algorithm applied to non-constant envelope signals according to comparative example 1;
fig. 5 is a graph comparing the change in NRMSE for different roll-off factors for the symbol rate estimation algorithm provided in comparative example 1 and the symbol rate estimation algorithm provided in example 2.
Detailed Description
The invention is further described below, but not limited thereto, with reference to the following examples and the accompanying drawings.
Example 1
An improved symbol rate estimation algorithm for a non-constant envelope signal for estimating the symbol rate of the non-constant envelope signal, as shown in fig. 1, the method comprising the steps of:
In step 1, the received signal sequence r (t) is obtained through an additive white gaussian noise channel, and the expression of r (t) is shown as formula (I):
r(t)＝s(t)+n(t) (I)
in the formula (I), n (t) represents a mean value of 0 and a variance of
White gaussian noise independent of the signal s (t), s (t) is expressed by the formula (II):
in the formula (II), x (t) represents a complex baseband symbol sequence after passing through a shaping filter, fcThe carrier frequency, x (t), is expressed as in formula (III):
in the formula (III), cn＝an+jbnRepresents a modulation symbol, anRepresenting the real part of the baseband modulation symbols, bnRepresenting the imaginary part of the baseband modulation symbols, n representing the number of baseband modulation symbols, gT(T) represents a function of the shaping filter, TbIs the symbol period.
The shaping filter is a root-raised cosine roll-off shaping filter. The root raised cosine roll-off shaping filter is selected to save frequency spectrum resources.
In the step 3, Rp(f) Is defined in the present invention as the relative projection, Rp(f) The calculation formula of (2) is as follows:
in the formula (IV), | U (f) | represents the amplitude value of the spectral line at the frequency f, | Ul(f)|、|Ur(f) I respectively represents the maximum value of the spectral line amplitude of the left k neighborhood of the frequency f; i Ur(f) I respectively represents the maximum value of the spectral line amplitude of the right k neighborhood of the frequency f; the left k neighborhood represents the frequency point range of [ f-k, f-1 ]]The right k neighborhood represents the frequency point range of [ f +1, f + k ]]The frequency spectrum of (a);
for lines in the continuum, their relative prominence R is due to their close proximity to the surrounding lines in magnitudepApproximately equal to 1, and the more prominent spectral lines are much larger than their surrounding spectral lines, with relative prominence RpFar greater than 1, the evaluation criteria presented herein can effectively evaluate the prominence of a spectral line with respect to its surrounding spectral lines. The extraction of the parameters can not only effectively depict the prominence of the code element rate spectral line, but also has better application significance in the modulation mode identification based on the spectral characteristics, and can numerically depict the spectral characteristics of signals.
In step 5, a coarse estimation F of the symbol rate is selectedcoaNear narrow band range [ (1-epsilon) Fcoa,(1+ε)Fcoa]In a narrow-band range, a simplified ZoomFFT is adopted to perform frequency selection analysis on { A (t) } to obtain a frequency spectrum M (f), and the specific steps are as follows:
5-1, selecting a thinning multiple D, wherein the size of the D determines the estimation accuracy, and the larger the D is, the higher the estimation accuracy is, but the corresponding calculated amount is increased, so that the relationship between the accuracy and the calculated amount is balanced;
5-2, mixing U+In the case of' (F) f.epsilon. [ F ]coa,(1+ε)Fcoa]The corresponding data is arranged and moved to the U in the forward direction+To the right of the zero frequency position of' (f), similarly, U+' (F) wherein F is within [ (1-epsilon) Fcoa,Fcoa) The corresponding data is arranged and moved to the U in the forward direction+Left side of zero frequency position of' (f), setting other position frequency spectrum data to zero to obtain new sequence S (f);
5-3, performing N/2-point inverse Fourier transform on the S (f) to obtain a time domain signal, wherein the value of N is more than or equal to the minimum power number of 2 of the length of the { A (t) } sequence;
5-4, resampling the time domain signal, wherein the sampling frequency of resampling is fs/D，fsRepresenting sampling frequency, namely taking data every other (D-1) points to obtain m (t), wherein m (t) represents a time domain signal after resampling;
5-5, performing N/2-point Fourier transform on m (t) to obtain a refined frequency spectrum M (f).
According to the invention, the envelope square spectrum of the signal is calculated, the relative saliency is calculated for the amplitude of each frequency point, the relative saliency is multiplied by the original amplitude to correct the envelope square spectrum, the maximum value of the corrected spectrum is extracted to obtain the rough estimation, and the frequency selection analysis is adopted to carry out the fine estimation, so that the estimated value of the code element rate is obtained. The improved algorithm solves the problem that the original algorithm has large estimation error in MASK signals and low roll-off factor forming signals, has wider signal application range than the original algorithm, has low complexity and is beneficial to engineering realization.
The impact of different modulation types on improving the performance of the algorithm was studied:
in order to verify the universality of the algorithm improvement algorithm on the modulation type, experiments are respectively carried out on {2ASK, 4ASK, 8ASK, BPSK, QPSK, 8PSK, 16PSK, pi/4 DQPSK, 8QAM, 16QAM, 32QAM and 64QAM } signals, the normalized symbol rate, the carrier frequency offset and the sampling rate are respectively set to be 1, 0.05 and 8, the number N of symbols is 1000, the normalized symbol rate, the carrier frequency offset and the sampling rate are respectively formed by a root lifting filter with a roll-off factor of 0.5, Gaussian white noise is superposed, the change range of the signal-to-noise ratio is 0-30dB, and the stepping is 1 dB. 100 experiments were performed at each signal-to-noise ratio and each modulation type.
In this embodiment, in step 3, k is selected to be 32. As shown in fig. 2(b), after the envelope square spectrogram of the 2ASK signal is processed in step 3, the symbol rate spectral line becomes a global maximum, and the symbol rate can be estimated well. As can be seen from fig. 2(a), the 2ASK signal has a symbol rate spectrum after the envelope square spectrum is processed by an unmodified algorithm, and the symbol rate spectrum is not a global maximum spectrum, which results in failure of the algorithm.
In the step 5, the thinning multiple D is 64, and the epsilon is 0.01. The refined spectrum M (F) processed in step 5 is shown in FIG. 2(c), and the frequency value F corresponding to the maximum value of the normalized amplitude of the spectrum M (F)accI.e. the symbol rate RbIs determined.
The estimated performance of the improved algorithm provided by the present invention is measured using Normalized Root Mean Square Error (NRMSE), as shown in equation (V):
formula (A), (B) andv), X represents an actual value,
the estimated value of the i-th experiment is shown, and M represents the total experiment times.
The normalized root mean square error of the above experiment is shown in FIG. 3(a) and FIG. 3(b), and it can be seen from FIG. 3(a) that the improved algorithm provided by the present invention performs well in MASK class, and the error is reduced to 10 when the SNR is greater than 4dB-4The following. As can be seen from FIG. 3(b), the improved algorithm provided by the present invention also performs well in MPSK and MQAM signals, and the error is reduced to 10 when the signal-to-noise ratio is greater than 4dB-4The following.
Comparative example 1
A symbol rate estimation algorithm for a non-constant envelope signal for estimating a symbol rate of the non-constant envelope signal, the method comprising the steps of:
When the simulation experiment parameters are the same as those in the embodiment 1, and the 2ASK, 4ASK and 8ASK signals are tested, the Normalized Root Mean Square Error (NRMSE) of the algorithm provided in the comparative example 1 is shown in fig. 4, and it can be seen that the estimation error of the algorithm is large in the three modulation modes, and the estimation error of the 2ASK and 4ASK is about 0.85; whileThe estimation error of the 8ASK signal is between 0.4 and 0.85; as can be seen from fig. 3(a) of the improved algorithm in embodiment 1, the Normalized Root Mean Square Error (NRMSE) of the improved algorithm provided by the present invention is reduced to 10 over 4dB for the 2ASK, 4ASK and 8ASK signals without filtering and noise reduction processing-4The following are preferable in applicability and noise resistance.
The influence of the roll-off factor on the symbol rate estimation performance is studied:
the normalized root mean square error of the algorithm provided in comparative example 1 and the improved algorithm provided by the present invention was calculated at different roll-off factors. The experimental parameters were set as follows: selecting a signal modulation mode of QPSK, wherein the normalized code element rate, the carrier frequency offset and the sampling rate are respectively 1, 0.05 and 8, the number N of code elements is 1000, the noise is white Gaussian noise, the signal to noise ratio is 10dB, selecting a roll-off factor change range [0.1,1] commonly used in a communication system, and stepping to 0.05; k was chosen to be 32, ε was 0.01, and 100 simulations were run for each roll-off factor.
As shown in fig. 5, when the roll-off factor is lower than 0.2, the estimation error of the algorithm provided in comparative example 1 is larger, while the improved algorithm provided by the present invention has better universality for the roll-off factor, and the mean square error is 10-5Order of magnitude, the improved algorithm performance provided by the present invention is not affected by the roll-off factor.
Claims (6)
1. An improved symbol rate estimation algorithm for a non-constant envelope signal, for estimating the symbol rate of the non-constant envelope signal, the method comprising the steps of:
step 1, squaring and transforming the envelope of the received signal sequence r (t) to obtain a new sequence { a (t) }, { a (t) } { | r (t) }2}；
Step 2, performing fast Fourier transform on the new sequence { A (t) } to obtain a frequency spectrum U (f), and taking the frequency spectrum U of the positive frequency domain part of the frequency spectrum U (f)+(f) And removing the direct current component;
step 3, traversing U+(f) Calculate U+(f) Relative projection R of each point inp(f) And is combined with U+(f) Multiplying to obtain corrected U+' (f), i.e. U+′(f)＝U+(f)·Rp(f)；
Step 4, searching U+' (F) the frequency value F corresponding to the maximum value of the normalized amplitudecoa，FcoaIs a coarse estimate of the symbol rate;
step 5, selecting a rough estimation value F of the code element ratecoaNarrow band range of [ (1-epsilon) F)coa,(1+ε)Fcoa]The parameter epsilon is used for controlling the narrow-band estimation range, and epsilon is more than 0 and less than 1; within a narrow band range, frequency selection analysis is carried out on { A (t) } to obtain a frequency spectrum M (F), and a frequency value F corresponding to the maximum value of the normalized amplitude of the frequency spectrum M (F)accI.e. the symbol rate RbIs determined.
2. The improved symbol rate estimation algorithm for non-constant envelope signals according to claim 1, wherein in step 1, the received signal sequence r (t) is obtained through an additive white gaussian noise channel, and the expression of r (t) is shown in formula (I):
r(t)＝s(t)+n(t) (I)
in the formula (I), n (t) represents a mean value of 0 and a variance of
White gaussian noise independent of the signal s (t), s (t) is expressed by the formula (II):
in the formula (II), x (t) represents a complex baseband symbol sequence after passing through a shaping filter, fcThe carrier frequency, x (t), is expressed as in formula (III):
in the formula (III), cn＝an+jbnRepresents a modulation symbol, anRepresentation of baseband modulation symbolsSection bnRepresenting the imaginary part of the baseband modulation symbols, n representing the number of baseband modulation symbols, gT(T) represents a function of the shaping filter, TbIs the symbol period.
3. The improved symbol rate estimation algorithm for a non-constant envelope signal provided in claim 2 wherein the shaping filter is a root-raised cosine roll-off shaping filter.
4. The improved symbol rate estimation algorithm for non-constant envelope signals as claimed in claim 1 wherein in step 3, Rp(f) The calculation formula of (2) is as follows:
in the formula (IV), | U (f) | represents the amplitude value of the spectral line at the frequency f, | Ul(f)|、|Ur(f) I respectively represents the maximum value of the spectral line amplitude of the left k neighborhood of the frequency f; i Ur(f) I respectively represents the maximum value of the spectral line amplitude of the right k neighborhood of the frequency f; the left k neighborhood represents the frequency point range of [ f-k, f-1 ]]The right k neighborhood represents the frequency point range of [ f +1, f + k ]]Of the spectrum of (c).
5. The improved symbol rate estimation algorithm for a non-constant envelope signal provided in claim 4 wherein k is 32.
6. The improved symbol rate estimation algorithm applicable to non-constant envelope signals according to claim 1, wherein in step 5, in a narrow-band range, a simplified zoomft is used to perform frequency-selecting analysis on { a (t) } to obtain a spectrum m (f), and the specific steps are as follows:
5-1, selecting a thinning multiple D, wherein the size of D determines the estimation accuracy, the larger D is, the higher the estimation accuracy is, but the corresponding calculated amount is increased, and the D is 32 or 64 in the relation of the balance accuracy and the calculated amount;
5-2, mixing U+In the case of' (F) f.epsilon. [ F ]coa,(1+ε)Fcoa]The corresponding data is arranged and moved to the U in the forward direction+To the right of the zero frequency position of' (f), similarly, U+' (F) wherein F is within [ (1-epsilon) Fcoa,Fcoa) The corresponding data is arranged and moved to the U in the forward direction+Left side of zero frequency position of' (f), setting other position frequency spectrum data to zero to obtain new sequence S (f);
5-3, performing N/2-point inverse Fourier transform on the S (f) to obtain a time domain signal, wherein the value of N is more than or equal to the minimum power number of 2 of the length of the { A (t) } sequence;
5-4, resampling the time domain signal, wherein the sampling frequency of resampling is fs/D，fsRepresenting sampling frequency, namely taking data every other (D-1) points to obtain m (t), wherein m (t) represents a time domain signal after resampling;
5-5, performing N/2-point Fourier transform on m (t) to obtain a refined frequency spectrum M (f).
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| CN114584444A (en) * | 2022-02-28 | 2022-06-03 | 哈尔滨工程大学 | Multi-h CPM modulation index estimation method based on cyclic moment characteristics |
| CN114584445A (en) * | 2022-03-08 | 2022-06-03 | 上海季丰电子股份有限公司 | Signal frequency estimation method, device, electronic equipment and medium |
| CN116962123A (en) * | 2023-09-20 | 2023-10-27 | 大尧信息科技(湖南)有限公司 | Raised cosine shaping filter bandwidth estimation method and system of software defined framework |
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